* Copyright (c) 1985 Regents of the University of California.
* Use and reproduction of this software are granted in accordance with
* the terms and conditions specified in the Berkeley Software License
* Agreement (in particular, this entails acknowledgement of the programs'
* source, and inclusion of this notice) with the additional understanding
* that all recipients should regard themselves as participants in an
* ongoing research project and hence should feel obligated to report
* their experiences (good or bad) with these elementary function codes,
* using "sendbug 4bsd-bugs@BERKELEY", to the authors.
"@(#)log.c 4.5 (Berkeley) 8/21/85; 1.3 (ucb.elefunt) %G%";
* RETURN THE LOGARITHM OF x
* DOUBLE PRECISION (VAX D FORMAT 56 bits or IEEE DOUBLE 53 BITS)
* CODED IN C BY K.C. NG, 1/19/85;
* REVISED BY K.C. NG on 2/7/85, 3/7/85, 3/24/85, 4/16/85.
* Required system supported functions:
* Required kernel function:
* 1. Argument Reduction: find k and f such that
* where sqrt(2)/2 < 1+f < sqrt(2) .
* 2. Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
* = 2s + 2/3 s**3 + 2/5 s**5 + .....,
* log(1+f) is computed by
* log(1+f) = 2s + s*log__L(s*s)
* log__L(z) = z*(L1 + z*(L2 + z*(... (L6 + z*L7)...)))
* See log__L() for the values of the coefficients.
* 3. Finally, log(x) = k*ln2 + log(1+f). (Here n*ln2 will be stored
* in two floating point number: n*ln2hi + n*ln2lo, n*ln2hi is exact
* since the last 20 bits of ln2hi is 0.)
* log(x) is NaN with signal if x < 0 (including -INF) ;
* log(+INF) is +INF; log(0) is -INF with signal;
* log(NaN) is that NaN with no signal.
* log(x) returns the exact log(x) nearly rounded. In a test run with
* 1,536,000 random arguments on a VAX, the maximum observed error was
* .826 ulps (units in the last place).
* The hexadecimal values are the intended ones for the following constants.
* The decimal values may be used, provided that the compiler will convert
* from decimal to binary accurately enough to produce the hexadecimal values
#ifdef VAX /* VAX D format */
/* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */
/* ln2lo = 1.6465949582897081279E-12 , Hex 2^-39 * .E7BCD5E4F1D9CC */
/* sqrt2 = 1.4142135623730950622E0 ; Hex 2^ 1 * .B504F333F9DE65 */
static long ln2hix
[] = { 0x72174031, 0x0000f7d0};
static long ln2lox
[] = { 0xbcd52ce7, 0xd9cce4f1};
static long sqrt2x
[] = { 0x04f340b5, 0xde6533f9};
#define ln2hi (*(double*)ln2hix)
#define ln2lo (*(double*)ln2lox)
#define sqrt2 (*(double*)sqrt2x)
ln2hi
= 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */
ln2lo
= 1.9082149292705877000E-10 , /*Hex 2^-33 * 1.A39EF35793C76 */
sqrt2
= 1.4142135623730951455E0
; /*Hex 2^ 0 * 1.6A09E667F3BCD */
static double zero
=0.0, negone
= -1.0, half
=1.0/2.0;
double logb(),scalb(),copysign(),log__L(),s
,z
,t
;
if(x
!=x
) return(x
); /* x is NaN */
k
=logb(x
); x
=scalb(x
,-k
);
if(k
== -1022) /* subnormal no. */
{n
=logb(x
); x
=scalb(x
,-n
); k
+=n
;}
if(x
>= sqrt2
) {k
+= 1; x
*= half
;}
z
=k
*ln2lo
+s
*(t
+log__L(s
*s
));
/* end of if (x > zero) */
return (infnan(-ERANGE
)); /* -INF */
return (infnan(EDOM
)); /* NaN */
/* zero argument, return -INF with signal */
/* negative argument, return NaN with signal */
/* end of if (finite(x)) */
/* NOT REACHED ifdef VAX */
/* log(-INF) is NaN with signal */